Large optical nonlinearity of ITO nanorods for sub-picosecond all-optical modulation of the full-visible spectrum

Nonlinear optical responses of materials play a vital role for the development of active nanophotonic and plasmonic devices. Optical nonlinearity induced by intense optical excitation of mobile electrons in metallic nanostructures can provide large-amplitude, dynamic tuning of their electromagnetic response, which is potentially useful for all-optical processing of information and dynamic beam control. Here we report on the sub-picosecond optical nonlinearity of indium tin oxide nanorod arrays (ITO-NRAs) following intraband, on-plasmon-resonance optical pumping, which enables modulation of the full-visible spectrum with large absolute change of transmission, favourable spectral tunability and beam-steering capability. Furthermore, we observe a transient response in the microsecond regime associated with slow lattice cooling, which arises from the large aspect-ratio and low thermal conductivity of ITO-NRAs. Our results demonstrate that all-optical control of light can be achieved by using heavily doped wide-bandgap semiconductors in their transparent regime with speed faster than that of noble metals.

shown in Supplementary Fig. 8d. The experimental data was acquired at 55° using a J. A.
Woollam M2000U instrument. b, Experimental and simulated Ψ and Δ angles for the ITO film in the NIR range; the data fitting assumed a Drude permittivity with ɛ ∞ = 3.90, ω p = 2.10 eV, and γ p = 0.0.065 eV. The experimental data was acquired at 70° using a Horiba UVISEL instrument. Pump fluences for a and b are 1.31 mJ·cm -2 and 9.2 mJ·cm -2 , respectively.

Supplementary Note 1: Calculation of the grating order intensities
The electromagnetic waves scattered by a periodic phased array can be decomposed into orthogonal eigenmodes, which are essentially the grating orders including both propagating and evanescent ones. Since the nanorod spacing of 1 µm is comparable to the wavelength in the visible range, higher order propagating modes (besides the zero order mode) can be produced. To extract intensities of these higher order modes from optical simulations, we decomposed the  are the electric field intensities for the (n, m) order. S = a 2 is the cross-sectional area of a unit cell, k xn = k x0 -2πn/a and k ym = k y0 -2πm/a. Here k x0 = k 0 ·sinθ·cosφ, k y0 = k 0 ·sinθ·sinφ, k 0 = 2π/λ is the incident wave vector, θ = 0 is the incident angle and φ = 0 is the azimuthal angle (the incident wave vector is normal to the substrate). The wave vector in the z direction is k znm = (k 0 2 -k xn 2k ym 2 ) 1/2 ; a mode is propagating when k znm is real and evanescent when k znm is imaginary.
Transmission of the (n, m) grating order is calculated as   To further verify that the wave propagating along the nanorod follows the fundamental HE 11 mode, we plot in Supplementary Fig. 1b to Fig. 1f the distributions of the electric field intensity averaged along the length of the nanorod at the wavelengths of the five transmission minima, which are similar to those reported for the HE 11 mode elsewhere 3, 4 , thereby justifying our use of the effective mode index for the estimation of the spectral locations of the transmission minima.

Supplementary Note 2: Permittivity of ITO
The static transmission spectrum of the ITO-NRA from 360 nm to 710 nm was fitted using Supplementary Fig. 6 shows the schematic diagram of the considered optical transition and various quantities defined. We denote R=m c /m v , and1/m r =1/m v +1/m c , where m r is the reduced effective mass. When an incident photon with energy ħω is absorbed in a direct, interband optical transition, energy conservation dictates that If we let k 2 = 2m c (E c +CE c 2 )/ħ 2 , equation (1) becomes ħω = E g +E c +R(E c +CE c 2 ). This is a quadratic equation in E c and can be rewritten as RCE c 2 +(R+1)E c +(E g -ħω) = 0, with the solution The derivative of E c with respective to the photon energy ħω is given by

(5).
Since α(ω) = 2ωn′′(ω)/c, we can write where n′′(ω) is the imaginary part of the refractive index. Combining equation ( which is a dimensionless quantity. Now the intraband optical pumping in our study gives rise to a redistribution of the electrons in the conduction band, whose temperature T can be calculated based on our earlier study 5 . As the Fermi function term f(E c ) in equation (7)   . The procedure described above was used as a model to theoretically calculate the change of real part of the relative permittivity (shown in Fig. 3f) due to the modification of the interband transition under intraband optical pumping, which is the case of the fast component in our TA experiments. Supplementary Fig. 7 presents the calculated f(E c , T), Δf(E c , T), E c (ħω), and Δf(ħω, T) for electron temperature ranging from 300 K to 14,500 K. The Δf(ħω, T) was introduced in Supplementary Fig. 7d to better illustrate the connection between Δf(E c , T) and Δε′′(ω, T). Note that at high electron temperatures (> 10,500 K), the electron chemical potential µ falls below the CBM, which is a result of conservation of the electron density.  Supplementary Fig. 8d). As it was not possible to get reliable absorption versus wavelength data for the ITO-NRA (arising from the scattering as well as absorption due to the substrate, which is evident from Supplementary Fig. 4), we performed ellipsometry measurements (in both the ultraviolet to visible, and near-infrared ranges) on an epitaxial ITO film (135 nm thick) sputtered on YSZ, with the results shown in Supplementary Fig. 9. The experimental α(ω) curve ( Supplementary Fig. 8d) was calculated using the refractive index obtained from the ellipsometric data fitting. The film plasma frequency was determined to be ~ 2.1 eV based on the ellipsometry data in the NIR (Supplementary Fig. 9b); this value is close to the deduced plasma frequency of 2.02 eV for the ITO-NRAs.

Supplementary Note 5: Estimating the electron and lattice temperatures
Measurements of the pump power showed that nearly 50% is transmitted and about 5% is The electron temperature at t e,0 is denoted as T e,0 . This was estimated using the procedure described earlier 5 . To assess the lattice temperature (denoted as T l,0 ) achieved at t l,0 , we used the heat capacity data from E. H. P. Cordfunke et al 1 for In 2 O 3 measured for the range from 0 to 1000 K. To convert this data into the required units we used the In 2 O 3 molecular weight of 277.64 g·mol -1 and a mass density of 7.16×10 3 kg·m -3 (calculated from the lattice constant of cubic ITO, 1.01 nm). For comparison purposes, Cordfunke's heat capacity is equivalent to 2.567×10 6 J·m -3 ·K -1 at 298 K, which is to be compared with a value of 2.58×10 6 J·m -3 ·K -1 adopted in the independent work by T. Yagi et al 15 .
Supplementary Fig. 10a and Fig. 10b show the calculated dependences of T e,0 and T l,0 on the pump fluence (shown as curves). Based on the experimental fluences we can estimate temperatures reached in the TA experiments (shown as circles). Notably, the initial temperature of electrons (maximum is ~ 14,000 K) is about two orders of magnitude higher than that of the lattice (maximum is ~ 500 K), which stems from their very different heat capacities, as shown in Supplementary Fig. 10c and Fig. 10d.

Supplementary Note 6: Details of the heat-transfer simulations
The heat transfer equation is given by ρC p (∂T L /∂t) + ∇·(-κ∇T L ) = 0 where the temperature T L is a function of both time and position, and κ is the thermal conductivity. This equation was solved using COMSOL Multiphysics in the time domain. A uniform temperature profile in the nanorod was used as the initial condition (with temperatures obtained from Supplementary Fig.   10b). Periodic boundary conditions were used along the in-plane directions. As no perfectly matched layer (which is an absorbing boundary) is available in the time domain study, we truncated the YSZ substrate in the out-of-plane direction at 10 µm below the interface of YSZ and ITO. A constant temperature (300 K) was imposed on the bottom YSZ boundary; this is valid since the total volume of YSZ is more than two orders of magnitude larger than the nanorod in the heat-transfer simulations, therefore temperature rise at the bottom boundary is at most a few degrees.
The thermal conductivity κ of ITO was calculated from the equation κ = κ el + κ ph , where κ el and κ ph are thermal conductivities contributed by mobile electrons and phonons, respectively.
According to T. Ashida et al 16 , κ ph is almost constant (3.95 W·m -1 ·K -1 for ITO films with different electron concentrations), whereas κ el is well described by the Wiedemann-Franz law of κ el = LTσ, where L is the Lorentz number (2.45×10 -8 WΩ·K -2 ) and σ is the electrical conductivity.
In our heat-transfer simulations we considered κ el = LTσ as a temperature dependent quantity, as opposed to κ ph which was assumed to be temperature independent. To get a reasonable estimate for σ, we performed Hall measurement (Van der Pauw method, Ecopia HMS-5000) on an epitaxial ITO film grown on YSZ substrate, whose electron concentration and mobility were found to be ~ 1.3×10 21 cm -3 and 47 cm 2 ·V -1 ·s -1 , respectively, yielding a value of 9.4881×10 5 S·m -1 for σ. The thermal conductivity of ITO at 300 K is determined to be 10.9 W·m -1 ·K -1 , which is more than an order of magnitude smaller than that of gold (314 W·m -1 ·K -1 ). The thermal conductivity and heat capacity of YSZ were taken to be 2.5 W·m -1 ·K -1 (from K. W. Schlichting et al 17 ) and 60.4 J·mol -1 ·K -1 (from T. Tojo et al 18 ), respectively. Both quantities were assumed to be temperature independent, since the temperature rise in the YSZ substrate is negligible in comparison to that of ITO. YSZ's molecular weight and mass density were 123.218 g·mol -1 and 6.0 g·cm -3 , respectively 17 . To further explore the geometrical dependence of the lattice heat dissipation rate, we performed extra heat-transfer simulations for ITO nanorods with different heights and edge lengths; the results are summarized in Supplementary Fig. 12. Interestingly, by adjusting the nanorod height the decay rate can be tuned over an order of magnitude. In contrast, changing the nanorod edge length has a negligible influence on the heat dissipation rate.

Supplementary Note 7: Full ∆T(t)/T(0) spectral maps
Supplementary Fig. 14 shows the ∆T(t)/T(0) spectral maps of the fast component acquired from short-delay TA experiments. The transient spectra of ∆T(t)/T(0) plotted in Fig. 2b are linecuts from these maps at time delay time t e,0 . Supplementary Fig. 15 presents the ∆T(t)/T(0) spectral maps of the slow component obtained from the long-delay TA experiments; the kinetics shown in Fig. 5b are line-cuts from these maps at 560 nm. A low signal-to-noise ratio below 400 nm arises from a relatively weak probe intensity.

Supplementary Note 8: Spectral maps of ∆OD(t) and T(t)
In TA experiments  Supplementary Fig. 15a for comparison with the corresponding ∆T(t)/T(0) spectral map (Fig. 2a). In addition, from T(0) and ∆T(t)/T(0) we can calculate T(t), which is a direct way to present the dynamic transmission property of the array. In Supplementary Fig. 15b we plot T(t) to demonstrate that the visible transmission spectrum first redshifts and then recovers rapidly in sub-picosecond time scales.